RD Sharma solution class 8 chapter 7 Factorization Exercise 7.6

Exercise 7.6

Page-7.22

Question 1:

Factorize each of the following algebraic expression:
4x² + 12xy +9y²

Answer 1:

$$\begin{gathered} 4x^2+12xy+9y^2 \\ =(2x{)}^2+2\times 2x\times 3y+(3y{)}^2 \\ =(2x+3y{)}^2 \\ =(2x+3y\left)\right(2x+3y) \end{gathered}$$

Question 2:

Factorize each of the following algebraic expression:
9a² − 24ab + 16b²

Answer 2:

$$\begin{gathered} 9a^2-24ab+16b^2 \\ =(3a{)}^2-2\times 3a\times 4b+(4b{)}^2 \\ =(3a-4b{)}^2 \\ =(3a-4b\left)\right(3a-4b) \end{gathered}$$

Question 3:

Factorize each of the following algebraic expression:
p²q² − 6pqr + 9r²

Answer 3:

$$\begin{gathered} p^2{q}^2-6pqr+9r^2 \\ =(pq{)}^2-2\times pq\times 3r+(3r{)}^2 \\ =(pq-3r{)}^2 \\ =(pq-3r\left)\right(pq-3r) \end{gathered}$$

Question 4:

Factorize each of the following algebraic expression:
36a² + 36a + 9

Answer 4:

$$\begin{gathered} 36{\mathrm{a}}^2+36\mathrm{a}+9 \\ =9(4{\mathrm{a}}^2+4\mathrm{a}+1) \\ =9\left\{(2\mathrm{a}{)}^2+2\times 2\mathrm{a}\times 1+1^2\right\} \\ =9(2\mathrm{a}+1{)}^2 \\ =9(2\mathrm{a}+1\left)\right(2\mathrm{a}+1) \end{gathered}$$

Page-7.23

Question 5:

Factorize each of the following algebraic expression:
a² + 2ab + b² − 16

Answer 5:

$$\begin{gathered} a^2+2ab+b^2-16 \\ =a^2+2\times a\times b+b^2-16 \\ =(a+b{)}^2-4^2 \\ =(a+b-4\left)\right(a+b+4) \end{gathered}$$

Question 6:

Factorize each of the following algebraic expression:
9z² − x² + 4xy − 4y²

Answer 6:

$$\begin{gathered} 9z^2-x^2+4xy-4y^2 \\ =9z^2-(x^2-4xy+4y^2) \\ =9z^2-[x^2-2\times x\times 2y+(2y{)}^2] \\ =(3z{)}^2-(x-2y{)}^2 \\ =[3z-(x-2y)\left]\right[3z+(x-2y)] \\ =(3z-x+2y\left)\right(3z+x-2y) \\ =(x-2y+3z\left)\right(-x+2y+3z) \end{gathered}$$

Question 7:

Factorize each of the following algebraic expression:
9a⁴ − 24a²b² + 16b⁴ − 256

Answer 7:

$$\begin{gathered} 9a^4-24a^2{b}^2+16b^4-256 \\ =(9a^4-24a^2{b}^2+16b^4)-256 \\ =\left[\right(3a^2{)}^2-2\times 3a^2\times 4b^2+\left(4b^2{)}^2\right]-{16}^2 \\ =(3a^2-4b^2{)}^2-{16}^2 \\ =[(3a^2-4b^2)-16\left]\right[(3a^2-4b^2)+16] \\ =(3a^2-4b^2-16\left)\right(3a^2-4b^2+16) \end{gathered}$$

Question 8:

Factorize each of the following algebraic expression:
16 − a⁶ + 4a³b³ − 4b⁶

Answer 8:

$$\begin{gathered} 16-a^6+4a^3{b}^3-4b^6 \\ =16-(a^6-4a^3{b}^3+4b^6) \\ =4^2-\left[\right(a^3{)}^2-2\times a^3\times 2b^3+\left(2b^3{)}^2\right] \\ =4^2-(a^3-2b^3{)}^2 \\ =[4-(a^3-2b^3)\left]\right[4+(a^3-2b^3)] \\ =(4-a^3+2b^3\left)\right(4+a^3-2b^3) \\ =(a^3-2b^3+4\left)\right(-a^3+2b^3+4) \end{gathered}$$

Question 9:

Factorize each of the following algebraic expression:
a² − 2ab + b² − c²

Answer 9:

$$\begin{gathered} a^2-2ab+b^2-c^2 \\ =(a^2-2ab+b^2)-c^2 \\ =(a^2-2\times a\times b+b^2)-c^2 \\ =(a-b{)}^2-c^2 \\ =[(a-b)-c\left]\right[(a-b)+c] \\ =(a-b-c\left)\right(a-b+c) \end{gathered}$$

Question 10:

Factorize each of the following algebraic expression:
x² + 2x + 1 − 9y²

Answer 10:

$$\begin{gathered} x^2+2x+1-9y^2 \\ =(x^2+2x+1)-9y^2 \\ =(x^2+2\times x\times 1+1)-9y^2 \\ =(x+1{)}^2-(3y{)}^2 \\ =\left[\right(x+1)-3y]\left[\right(x+1)+3y] \\ =(x+1-3y)(x+1+3y) \\ =(x+3y+1)(x-3y+1) \end{gathered}$$

Question 11:

Factorize each of the following algebraic expression:
a² + 4ab + 3b²

Answer 11:

$$\begin{gathered} {\mathrm{a}}^2+4\mathrm{ab}+3{\mathrm{b}}^2 \\ ={\mathrm{a}}^2+4\mathrm{ab}+4{\mathrm{b}}^2-{\mathrm{b}}^2 \\ =[{\mathrm{a}}^2+2\times \mathrm{a}\times 2\mathrm{b}+(2\mathrm{b}{)}^2]-{\mathrm{b}}^2 \\ =(\mathrm{a}+2\mathrm{b}{)}^2-{\mathrm{b}}^2 \\ =\left[\right(\mathrm{a}+2\mathrm{b})-\mathrm{b}]\left[\right(\mathrm{a}+2\mathrm{b})+\mathrm{b}] \\ =(\mathrm{a}+2\mathrm{b}-\mathrm{b})(\mathrm{a}+2\mathrm{b}+\mathrm{b}) \\ =(\mathrm{a}+\mathrm{b})(\mathrm{a}+3\mathrm{b}) \end{gathered}$$

Question 12:

Factorize each of the following algebraic expression:
96 − 4x − x²

Answer 12:

$$\begin{gathered} 96-4\mathrm{x}-{\mathrm{x}}^2 \\ =100-4-4\mathrm{x}-{\mathrm{x}}^2 \\ =100-({\mathrm{x}}^2+4\mathrm{x}+4) \\ =100-({\mathrm{x}}^2+2\times \mathrm{x}\times 2+2^2) \\ ={10}^2-(\mathrm{x}+2{)}^2 \\ =[10-(\mathrm{x}+2)\left]\right[10+(\mathrm{x}+2)] \\ =(10-\mathrm{x}-2\left)\right(10+\mathrm{x}+2) \\ =(8-\mathrm{x}\left)\right(12+\mathrm{x}) \\ =(x+12\left)\right(-x+8) \end{gathered}$$

Question 13:

Factorize each of the following algebraic expression:
a⁴ + 3a² +4

Answer 13:

$$\begin{gathered} {\mathrm{a}}^4+3{\mathrm{a}}^2+4 \\ ={\mathrm{a}}^4+4{\mathrm{a}}^2-{\mathrm{a}}^2+4 \\ =({\mathrm{a}}^4+4{\mathrm{a}}^2+4)-{\mathrm{a}}^2 \\ =\left[\right({\mathrm{a}}^2{)}^2+2\times {\mathrm{a}}^2\times 2+2^2]-{\mathrm{a}}^2 \\ =({\mathrm{a}}^2+2{)}^2-{\mathrm{a}}^2 \\ =\left[\right({\mathrm{a}}^2+2)-\mathrm{a}]\left[\right({\mathrm{a}}^2+2)+\mathrm{a}] \\ =({\mathrm{a}}^2-\mathrm{a}+2)\left(\right({\mathrm{a}}^2+\mathrm{a}+2) \end{gathered}$$

Question 14:

Factorize each of the following algebraic expression:
4x⁴ + 1

Answer 14:

$$\begin{gathered} 4{\mathrm{x}}^4+1 \\ =4{\mathrm{x}}^4+4{\mathrm{x}}^2+1-4{\mathrm{x}}^2 \\ =\left[\right(2{\mathrm{x}}^2{)}^2+2\times 2{\mathrm{x}}^2\times 1+1]-4{\mathrm{x}}^2 \\ =(2{\mathrm{x}}^2+1{)}^2-(2\mathrm{x}{)}^2 \\ =[(2{\mathrm{x}}^2+1)-2\mathrm{x}\left]\right[(2{\mathrm{x}}^2+1)+2\mathrm{x}] \\ =(2{\mathrm{x}}^2-2\mathrm{x}+1\left)\right(2{\mathrm{x}}^2+2\mathrm{x}+1) \end{gathered}$$

Question 15:

Factorize each of the following algebraic expression:
4x⁴ + y⁴

Answer 15:

$$\begin{gathered} 4{\mathrm{x}}^4+{\mathrm{y}}^4 \\ =4{\mathrm{x}}^4+4{\mathrm{x}}^2{\mathrm{y}}^2+{\mathrm{y}}^4-4{\mathrm{x}}^2{\mathrm{y}}^2 \\ =\left[\right(2{\mathrm{x}}^2{)}^2+2\times 2{\mathrm{x}}^2\times \mathrm{y}+\left({\mathrm{y}}^2{)}^2\right]-(2\mathrm{xy}{)}^2 \\ =(2{\mathrm{x}}^2+{\mathrm{y}}^2{)}^2-(2\mathrm{xy}{)}^2 \\ =[(2{\mathrm{x}}^2+{\mathrm{y}}^2)-2\mathrm{xy}\left]\right[(2{\mathrm{x}}^2+{\mathrm{y}}^2)+2\mathrm{xy}] \\ =(2{\mathrm{x}}^2-2\mathrm{xy}+{\mathrm{y}}^2\left)\right(2{\mathrm{x}}^2+2\mathrm{xy}+{\mathrm{y}}^2) \end{gathered}$$

Question 16:

Factorize each of the following algebraic expression:
(x + 2)² − 6(x + 2) + 9

Answer 16:

$$\begin{gathered} (\mathrm{x}+2{)}^2-6(\mathrm{x}+2)+9 \\ =(\mathrm{x}+2{)}^2-2\times (\mathrm{x}+2)\times 3+3^2 \\ =\left[\right(\mathrm{x}+2)-3{]}^2 \\ =(\mathrm{x}+2-3{)}^2 \\ =(\mathrm{x}-1{)}^2 \\ =(\mathrm{x}-1\left)\right(\mathrm{x}-1) \end{gathered}$$

Question 17:

Factorize each of the following algebraic expression:
25 − p² − q² − 2pq

Answer 17:

$$\begin{gathered} 25-{\mathrm{p}}^2-{\mathrm{q}}^2-2\mathrm{pq} \\ =25-({\mathrm{p}}^2+2\mathrm{pq}+{\mathrm{q}}^2) \\ =5^2-({\mathrm{p}}^2+2\times \mathrm{p}\times \mathrm{q}+{\mathrm{q}}^2) \\ =5^2-(\mathrm{p}+\mathrm{q}{)}^2 \\ =[5-(\mathrm{p}+\mathrm{q})\left]\right[5+(\mathrm{p}+\mathrm{q})] \\ =(5-\mathrm{p}-\mathrm{q}\left)\right(5+\mathrm{p}+\mathrm{q}) \\ =-(\mathrm{p}+\mathrm{q}-5\left)\right(\mathrm{p}+\mathrm{q}+5) \end{gathered}$$

Question 18:

Factorize each of the following algebraic expression:
x² + 9y² − 6xy − 25a²

Answer 18:

$$\begin{gathered} {\mathrm{x}}^2+9{\mathrm{y}}^2-6\mathrm{xy}-25{\mathrm{a}}^2 \\ =({\mathrm{x}}^2-6\mathrm{xy}+9{\mathrm{y}}^2)-25{\mathrm{a}}^2 \\ =[{\mathrm{x}}^2-2\times \mathrm{x}\times 3\mathrm{y}+(3\mathrm{y}{)}^2]-25{\mathrm{a}}^2 \\ =(\mathrm{x}-3\mathrm{y}{)}^2-(5\mathrm{a}{)}^2 \\ =[(\mathrm{x}-3\mathrm{y})-5\mathrm{a}\left]\right[(\mathrm{x}-3\mathrm{y})+5\mathrm{a}] \\ =(\mathrm{x}-3\mathrm{y}-5\mathrm{a}\left)\right(\mathrm{x}-3\mathrm{y}+5\mathrm{a}) \end{gathered}$$

Question 19:

Factorize each of the following algebraic expression:
49 − a² + 8ab − 16b²

Answer 19:

$$\begin{gathered} 49-{\mathrm{a}}^2+8\mathrm{ab}-16{\mathrm{b}}^2 \\ =49-({\mathrm{a}}^2-8\mathrm{ab}+16{\mathrm{b}}^2) \\ =49-[{\mathrm{a}}^2-2\times \mathrm{a}\times 4\mathrm{b}+(4\mathrm{b}{)}^2] \\ =7^2-(\mathrm{a}-4\mathrm{b}{)}^2 \\ =[7-(\mathrm{a}-4\mathrm{b}\left)\right][7+(\mathrm{a}-4\mathrm{b}\left)\right] \\ =(7-\mathrm{a}+4\mathrm{b})(7+\mathrm{a}-4\mathrm{b}) \\ =-(\mathrm{a}-4\mathrm{b}-7)(\mathrm{a}-4\mathrm{b}+7) \\ =-(\mathrm{a}-4\mathrm{b}+7)(\mathrm{a}-4\mathrm{b}-7) \end{gathered}$$

Question 20:

Factorize each of the following algebraic expression:
a² − 8ab + 16b² − 25c²

Answer 20:

$$\begin{gathered} a^2-8ab+16b^2-25c^2 \\ =(a^2-8ab+16b^2)-25c^2 \\ =[a^2-2\times a\times 4b+(4b{)}^2]-25c^2 \\ =(a-4b{)}^2-(5c{)}^2 \\ =[(a-4b)-5c\left]\right[(a-4b)+5c] \\ =(a-4b-5c\left)\right(a-4b+5c) \end{gathered}$$

Question 21:

Factorize each of the following algebraic expression:
x² − y² + 6y − 9

Answer 21:

$$\begin{gathered} x^2-y^2+6y-9 \\ =x^2-(y^2-6y+9) \\ =x^2-(y^2-2\times y\times 3+3^2) \\ =x^2-(y-3{)}^2 \\ =[x-(y-3)\left]\right[x+(y-3)] \\ =(x-y+3\left)\right(x+y-3) \end{gathered}$$

Question 22:

Factorize each of the following algebraic expression:
25x² − 10x + 1 − 36y²

Answer 22:

$$\begin{gathered} 25x^2-10x+1-36y^2 \\ =(25x^2-10x+1)-36y^2 \\ =\left[\right(5x{)}^2-2\times 5x\times 1+1]-36y^2 \\ =(5x-1{)}^2-(6y{)}^2 \\ =[(5x-1)-6y\left]\right[(5x-1)+6y] \\ =(5x-1-6y\left)\right(5x-1+6y) \\ =(5x-6y-1\left)\right(5x+6y-1) \end{gathered}$$

Question 23:

Factorize each of the following algebraic expression:
a² − b² + 2bc − c²

Answer 23:

$$\begin{gathered} a^2-b^2+2bc-c^2 \\ =a^2-(b^2-2bc+c^2) \\ =a^2-(b^2-2\times b\times c+c^2) \\ =a^2-(b-c{)}^2 \\ =[(a-(b-c\left)\right]\left[\right(a+(b-c)] \\ =(a-b+c\left)\right(a+b-c) \end{gathered}$$

Question 24:

Factorize each of the following algebraic expression:
a² + 2ab + b² − c²

Answer 24:

$$\begin{gathered} a^2+2ab+b^2-c^2 \\ =(a^2+2ab+b^2)-c^2 \\ =(a^2+2\times a\times b+b^2)-c^2 \\ =(a+b{)}^2-c^2 \\ =[(a+b)-c\left]\right[(a+b)+c] \\ =(a+b-c\left)\right(a+b+c) \end{gathered}$$

Question 25:

Factorize each of the following algebraic expression:
49 − x² − y² + 2xy

Answer 25:

$$\begin{gathered} 49-{\mathrm{x}}^2-{\mathrm{y}}^2+2\mathrm{xy} \\ =49-({\mathrm{x}}^2-2\mathrm{xy}+{\mathrm{y}}^2) \\ =49-({\mathrm{x}}^2-2\times \mathrm{x}\times \mathrm{y}+{\mathrm{y}}^2) \\ =7^2-(\mathrm{x}-\mathrm{y}{)}^2 \\ =[7-(\mathrm{x}-\mathrm{y})\left]\right[7+(\mathrm{x}-\mathrm{y})] \\ =(7-\mathrm{x}+\mathrm{y}\left)\right(7+\mathrm{x}-\mathrm{y}) \\ =(\mathrm{x}-\mathrm{y}+7\left)\right(\mathrm{y}-\mathrm{x}+7) \end{gathered}$$

Question 26:

Factorize each of the following algebraic expression:
a² + 4b² − 4ab − 4c²

Answer 26:

$$\begin{gathered} a^2+4b^2-4ab-4c^2 \\ =(a^2-4ab+4b^2)-4c^2 \\ =[a^2-2\times a\times 2b+(2b{)}^2]-4c^2 \\ =(a-2b{)}^2-(2c{)}^2 \\ =[(a-2b)-2c\left]\right[(a-2b)+2c] \\ =(a-2b-2c\left)\right(a-2b+2c) \end{gathered}$$

Question 27:

Factorize each of the following algebraic expression:
x² − y² − 4xz + 4z²

Answer 27:

$$\begin{gathered} {\mathrm{x}}^2-{\mathrm{y}}^2-4\mathrm{xz}+4{\mathrm{z}}^2 \\ =({\mathrm{x}}^2-4\mathrm{xz}+4{\mathrm{z}}^2)-{\mathrm{y}}^2 \\ =[{\mathrm{x}}^2-2\times \mathrm{x}\times 2\mathrm{z}+(2\mathrm{z}{)}^2]-{\mathrm{y}}^2 \\ =(\mathrm{x}-2\mathrm{z}{)}^2-{\mathrm{y}}^2 \\ =\left[\right(\mathrm{x}-2\mathrm{z})-\mathrm{y}]\left[\right(\mathrm{x}-2\mathrm{z})+\mathrm{y}] \\ =(\mathrm{x}-2\mathrm{z}-\mathrm{y})(\mathrm{x}-2\mathrm{z}+\mathrm{y}) \\ =(\mathrm{x}+\mathrm{y}-2\mathrm{z})(\mathrm{x}-\mathrm{y}-2\mathrm{z}) \end{gathered}$$